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Network Competition with Reciprocal

Proportional Access Charge Rules

Toker DOGANOGLU†and Yair TAUMAN‡

December 2, 1996

SUNY at Stony Brook Discussion Paper

DP96-01

We would like to thank Tom Muench, Pradeep Dubey and Paul Teske for helpful comments.

This is a very preliminary version of the paper and all errors are ours.†Department of Economics, SUNY at Stony Brook, Stony Brook, NY 11794,USA.‡Department of Economics, SUNY at Stony Brook, Stony Brook, NY 11794,USA and Faculty

of Management, Tel-Aviv University, ISRAEL.

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Abstract

This paper presents a model of two competing local telecommunicationsnetworks, similar in spirit to the model of Laffont, Rey and Tirole (1996).

The networks have different attributes which we assume are fixed and the

consumers have idiosyncratic tastes for these attributes. The networks are

mandated to interconnect and the access charges are determined coopera-

tively in the first stage. In the second stage, the two network companies are

engaged in a price competition to attract consumers. In the third stage, each

consumer selects a network and determines the consumption of phone calls.

Laffont, Rey and Tirole (1996) have shown that except for restrictive

scenarios, the local price competition does not result in a pure strategy equi-

librium. In this paper, we assume that the two companies choose access

charge rules rather than simply access charges. These rules determine theaccess charges as a function of the future local prices. We show that the

family of reciprocal proportional access charge rules generates a pure strat-

egy equilibrium and we discuss its properties. (JEL D4, K21, L41,43, L51,

L96)

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Introduction

The telecommunications industry has been one of the fastest developing industries

of the last half century. Traditionally, the telephone industry is considered to be a

natural monopoly, since the cost structure consists of a large fixed cost component

and a decreasing average operating cost (see Noam (1994)). For this reason, in

most countries the telecommunications companies are owned by the governments.

The inefficiencies generated by government ownership of a technologically very

sophisticated industry has led to privatization in several countries and this trend is

still in effect. In the US, while the telecommunications industry has always been

privately owned, it has been subject to substantial regulation.

Over the last thirty years, with the emergence of entrants in several segments

of the market, the question of regulating the telecommunications industry has be-come even more complex. The two well-known examples of such entrants are

Microwave Communications Inc., MCI, in the US and Mercury Communications

in the UK, which led to major policy changes in their respective countries. Both

of these companies have provided long distance service for the consumers using

local networks of the incumbent monopolists, AT&T in the US, and BT in the

UK. For the history of the evolution of competition in the telecommunications

industries in these two countries see Vogelsang and Mitchell (1994). In the US,

the AT&T monopoly was broken down into a long distance company and seven

Regional Bell Operating Companies (RBOCs) which were awarded monopolies

for local service in their operating areas. The motivation behind this was that a

vertically integrated monopoly would not have incentives to let a competitor enter

in some segment of their business. The RBOCs were not allowed to provide long

distance service, while MCI and AT&T were banned from local access markets.

The long distance companies had to pay the local network companies access fees

to interconnect with their networks and the fees were determined by the FCC. In

the UK, the monopoly stayed intact and an access charge mechanism was designed

for competitors to access BT’s local networks. OFTEL, the regulatory agency in

the UK, has set the rates for interconnection (see Laffont and Tirole (1994a)).

The access charge mechanism design has been a subject of immense discus-

sion in the last decade, most of it in the context of one vertically integrated firm

competing in one segment of the market. A competitor has to have access tothe other segment and access charges need to be determined. Laffont and Ti-

role (1994b) proposed a mechanism which yields the welfare maximizing access

charges. The Efficient Component Pricing Rule (ECPR) of Baumol and Sidak

(1994), despite its ease, generated a lot of discussion since it is efficient only un-

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der very strict assumptions, and seems to favor the incumbent monopolies. Econo-

mides and White (1995) present a critique of ECPR, while Armstrong Doyle andVickers (1995) reinterpret the ECPR in the light of Laffont and Tirole (1994b).

Economides and Woroch (1992) examine the incentives of two long distance com-

panies for interconnection when the bottleneck facility is owned only by one of

them. They calculated the final prices for several different scenarios as well as the

access charges.

In recent years, breakthrough technologies like the introduction of the fiber op-

tic lines, mobile communication networks, the transformation of cable networks

to carry phone calls and the amazing growth of Internet, have questioned the ne-

cessity of monopolies in the local access market (see Noam (1994)). The current

trend is to open the whole telecommunications market to competition. Several

countries are planning for a competitive telecommunications industry and passinglegislation to prepare the legal grounds for competition. The latest example of

such policies is the Telecommunications Act of 1996 in the US, which essentially

allows entry into the telecommunications market. These developments bring new

questions to mind concerning the new policies that will be required to accord with

the existence of several networks and their mutual interconnection.

One of the most important articles in the 1996 act is the one which addresses

interconnection between networks. It asserts that interconnection should be pro-

vided on a nondiscriminatory manner to everyone who wishes; the access to net-

works should be at a just and fair price; the access charges should be negotiated

between interacting firms and binding agreements should be signed. These agree-

ments are subject to the approval of FCC and Public Utility Commissions. 1 Like

most laws, the 1996 bill uses vague language and it is subject to interpretation.

Once the access fees are set by mutual consent, the networks act competitively,

i.e. they compete in local pricing schemes, service quality, etc.

Laffont, Rey and Tirole (1996) (hereafter LRT) have analyzed a model of two

local network companies that possess different attributes for consumers.2 In their

model, the two companies, given access charges, set the local prices competitively.

The customers of a network are charged the same price independent of the network

which completes their call. The networks compete only in prices since the other

attributes are assumed to be fixed. Then consumers select their preferred networks

a la Hotelling. It is shown in LRT that equilibrium in local prices may not existexcept for restrictive values of access charges.

1See Telecommunications Act of 1996 .2Carter and Wright (1996) build on LRT and examine the effects of brand loyalty.

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This paper deals with a similar model. It differs from LRT in two aspects:

the determination of access fees and the determination of consumer demands. Inthe first stage, the two companies negotiate access charge rules rather than access

charges. The access charges are functions of the local prices and they are deter-

mined only after the second stage, when the local prices are determined. Thus, any

change in demands and therefore in local prices will automatically result in new

access charges. The other difference between our model and that of LRT is in the

consumer preferences. The utilities of consumers in both models consist of a de-

terministic part and a random part. While the deterministic part in LRT generates

demands with constant price elasticity, we use quadratic utilities which generate

linear demands. This avoids the unboundedness of the consumption for small

prices and provides a satiation point, which is natural for this type of services. 3

For the random component we use the Weibull distribution, which resembles theNormal distribution and provides analytical convenience. Furthermore, it is easy

to extend this model to deal with more than two network companies.

It is shown here that if the network companies restrict their first stage negoti-

ation to reciprocal4 proportional access charge rules (RPACR), which determine

the access charges as certain proportions of the future local prices, then a pure

strategy equilibrium always exists and the local prices are explicitly computed. If

the companies are allowed to set the proportion factor to maximize their joint prof-

its then they will set access charges smaller than local prices when the networks

are not close substitutes. However, in this case the resulting local price coincides

with the monopoly price. In other words, the companies charge the customers the

monopoly price but charge each other a lower price.

If the two networks are close substitutes, they act competitively and the result

is low local prices even under joint profit maximization. In this case, the access

charges will coincide with the local prices. The adoption of RPACR can be viewed

as a mild and simple regulatory policy. It does not require information about the

determinants of the industry. If the services of the networks are close substi-

tutes then the use of RPACR results in competitive prices. In addition, as noted

above, RPACR generates access charges which are responsive to future changes

like demand shocks and technological innovations. However, if one believes that

the competing companies will find a way to increase differentiation then further

government intervention may be necessary.The paper is organized as follows. In Section 1, we introduce the general

3There is a limit to the time individuals spend on phone calls.4Reciprocal here means that both networks employ the same rule.

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model for the industry. In Section 2, we develop a model for consumer choice and

derive their demands. In Section 3, we analyze the competition between the twonetwork companies. Concluding remarks appear in Section 4.

1 The Industry Model

The telecommunications industry is a complicated industry to model. We deal

with a simple case of two network companies which only provide local telecom-

munications services. We refer to them as Network 1 and Network 2. The net-

works incur zero marginal cost for each call.

There is a fixed cost associated with building a network. Operating and main-

tenance costs are assumed to be independent of the amount of service provided.Usually, operating and maintenance costs do depend on the number of customers

of a network. But to simplify the analysis, we assume that these costs are in-

cluded in the fixed cost component.5 The fixed cost of each network is denoted

by F . Each company faces two demand functions. The demand function X 11, for

calls initiated and completed at Network 1, and the demand function X 12, for the

calls initiated at Network 1 and completed at Network 2. The demand functions

for Network 2, X 22 and X 21, are defined similarly. The demand functions X i j will

be derived from utility maximization. This is done in the next section.

The two companies first negotiate access charge rules, a1 and a2, where a1

is the per unit price that Network 2 will pay Network 1 for each unit call that is

completed in Network 1. The term a2 is similarly defined. Then they are engagedin a price competition which determines the local prices pk , (k

= 1; 2). The price

pk is the per unit charge of company k to each of its customers whether their calls

are completed locally or in the other network. In contrast to LRT, the access fees

may depend on the local prices p1 and p2. Thus, it is assumed that ak =

ak p1 ;

p2

for k =

1;

2. We further restrict our attention to the simple case of the proportional

rule

ak =

ak pk ;

where 0

ak

1, for k =

1;

2. If a1 =

a2 =

a, then the access charge rule will

be reciprocal, and it is called the Reciprocal Proportional Access Charge Rule

(RPACR). Thus, we treat each network as a regular customer who only buys par-tial service (just completion of a call) and therefore pays only a proportion of the

5Computer simulations suggest that our results will remain true without this assumption, at

least when these costs are not too large relative to the fixed cost. However the methods we use to

prove our result does not apply to this case.

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full service price. This is consistent with the nondiscrimination requirement of

the Telecommunications Act of 1996. The profit functions of the two networksare given by

Π1 =

p1 X 11 +

p1

ap2

X 12 +

ap1 X 21

F ;

(1)

Π2 =

p2 X 22 +

p2

ap1

X 21 +

ap2 X 12

F :

(2)

Consider the following sequence of events

Stage 1. The network companies select an RPACR by mutual consent.

Stage 2. The two companies choose their prices p1 and p2 simultaneously and

independently and announce them publicly.

Stage 3. After observing the prices p1 and p2, every consumer selects a net-

work to subscribe to and chooses the amount of calls.The Telecommunications Act of 1996 requires companies to negotiate and

then to sign binding agreements concerning their access charges before they en-

gage in the competition for local prices, services, etc. In our model, the parties

negotiate rules, not charges. The rules then determine access prices as functions of

their local prices. That is, the access charges are determined only after local prices

p1 and p2 are determined. Any change in local prices will have an immediate im-

pact on the access prices. A crucial point is to determine how to select access

charge rules in the bargaining stage. In our model it amounts to the selection of

the proportion a2 0 ; 1 .

2 Consumer Demand For Local Telecommunications

Services

In this section, we will specify the consumers’ utilities and model the process by

which they select their networks. Suppose that there are N potential consumers.

Each consumer is assumed to have idiosyncratic tastes which depend on the vari-

ous attributes of the networks (like specific services offered by the companies, the

intensity of advertising, accounting methods, etc.). This allows us to explain the

existence of several networks with similar products but different prices. Consumer

i who subscribes to network k has the following utility function,

ui

k ;

x;

y =

r

sx

x+

y

eσεik ;

(3)

where x is the total consumption of telephone calls in minutes and y is the amount

of numeraiare (money) consumed. The term εik measures the idiosyncratic taste

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variable of the consumer. It is assumed that εik ’s are Weibull distributed ((Domen-

cich and McFadden 1975)), statistically independent for all i and k , and that theyare private information of the consumers. The cdf of the Weibull distribution

closely resembles that of the Normal distribution. The term σ is a measure of the

dispersion of tastes , that is, σ measures the substitutability between the services

of the two networks. As σ!

0 the networks become perfect substitutes, while as

σ!

∞ the networks become perfect complements. One important feature of this

utility function is that the deterministic part will result in a linear demand function.

This implies a bounded amount of calls demanded at prices close to zero. Another

feature of the quadratic utility function is that it provides a satiation point, which

is natural for such services as there is a limit to the time an individual will spend

on the phone.

Random utility models of this kind have been extensively employed in the lit-erature, starting with Domencich and McFadden (1975); see Anderson, de Palma

and Thisse (1992) for a wide variety of examples.

Let V ik x be the deterministic part of the surplus of consumer i who subscribes

to Network k and consumes x units. Then

V ik x =

r

sx

x

pk x:

(4)

This is maximized for

xk =

1

2s

r

pk ;

(5)

and the maximum is given by

V ik =

1

4s

r

pk

2: (6)

Observe that (as in LRT) the demand of each consumer, xk does not depend on i;

thus we have dropped the index i.

By (3) and (6) consumer i prefers Network k to Network k if and only if

V ik eσεik

V ik eσεik

:

Therefore, the network companies assign probability Pik that i will choose Net-

work k over k , where

Pik =

Probability

V ik eσεik

V ik eσεik

:

(7)

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Since the εik ’s are independent and Weibull distributed, this probability is given

by (Anderson, de Palma and Thisse 1992)

Pik =

1

1+

V ik V ik

1σ

:

(8)

For the derivation of (8) see Anderson, de Palma and Thisse (1992).

Applying (6) and (8) to the case k=1, we have

Pi1 =

r

p1

τ

r

p1

τ+

r

p2

τ;

(9)

where τ =

2σ . Thus, this probability does not depend on i, and we can drop the

index i from Pi1 and write P1. From the point of view of the two companies, every

consumer will select Network 1 with probability P1. The expected number of consumers who will subscribe to Network 1 is therefore N P1. Notice that Pk can

be viewed as the market share of Network k . Consequently by (9), the expected

market share, m

p1 ;

p2

, of Network 1 is given by

m

p1 ;

p2 =

r

p1

τ

r

p1

τ+

r

p2

τ: (10)

The expected market share of Network 2 is obviously P2 =

1

m

p1 ;

p2

. Observe

that the aggregate subscriber demand faced by Network k is given by,

X k =

X k p1 ;

p2 =

N

2s

r

pk

Pk ; k =

1;

2:

(11)

Next let us find the expected number of calls initiated in Network k , and com-

pleted in Network j where k ;

j2 f 1 ; 2g . From the companies’ point of view, the

probability that a customer of Network k , k 2 f 1; 2g , calling a customer of Net-

work j is m

p1 ;

p2

if j=

1 and 1

m

p1 ;

p2

if j=

2. Therefore by (10) and

(11), we conclude that,

X 11 =

N

2s

r

p1

m

p1 ;

p2

2; (12)

X 12 =

N

2s

r

p1

m

p1 ;

p2 1

m

p1 ;

p2 ; (13)

X 22

=

N

2s

r

p2

m

p1

;

p2

2;

(14)

X 21 =

N

2s

r

p2

m

p1 ;

p2

1

m

p1 ;

p2 :

(15)

We use these demand functions in the next section to analyze the competition

between the two companies.

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3 Network Competition with RPACR

In this section, we analyze the price competition when the access charge rules are

reciprocal proportional rules. This means that both networks use the same rule,

ak =

apk ; 0

a

1;

k 2

1;

2, to calculate access charges. The number a is

determined in the first stage of the game. To simplify notation, we use p1 =

p and

p2 =

q. Using this rule we obtain, by (1), (2), and (12)-(15)

Π1 =

N

2sf

p

r

p

m

p;

q +

p

ap

r

p

m

p;

q

1

m

p;

q +

+

ap

r

q

m

p;

q 1

m

p;

q g

F ; (16)

Π2 =

N

2s

f

q

r

q 1

m

p;

q +

q

aq

r

q

m

p;

q 1

m

p;

q +

+

aq

r

p

m

p;

q

1

m

p;

q g

F :

(17)

Given a;

0

a

1, the two network companies are engaged in a price competi-

tion. We find that when N is sufficiently large to cover the fixed costs, a symmetric

equilibrium exists6 and can be computed explicitly.

Theorem 1: For every 0

a

1, τ

0 and r

0, there exists one and only one

symmetric equilibrium. It is given by:

p =

q =

2+

a

r

τ +

4

Proof. See Appendix.The main contribution of Theorem 1 is that pure strategy equilibrium in local

prices always exists with RPACR. The equilibrium prices have intuitive properties.

They are decreasing in substitutability rate, τ. The higher the substitutability rate

τ, the stronger is the competition between the networks, and, therefore, the lower

are the prices. Second, the prices are increasing in the demand intensity, r , in a

linear way. Finally they are increasing in the access charge proportion factor a.

The proportion a can be viewed as the marginal cost of a call that is completed in

the other network. The result asserts that the higher this cost is, the higher is the

price that the company charges to its customers.

The value of a is determined in the first stage by negotiation. The bargain-ing between the two parties may result in essentially any number in

0;

1

with

the restriction that the revenue of a company covers the fixed cost. Let us first

6Although we did not succeed in proving the uniqueness of this equilibrium, computer simu-

lations suggest that it is indeed unique.

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examine the case where the companies set a to maximize total profit, ΠT . Since

m

p ;

p =

1=

2, by Theorem 1,

ΠT = Π1 + Π2 =

N

2s

τ +

2

a

2+

a

r 2

τ + 4

2 2F

: (18)

This function is concave in a and it obtains its unconstrained maximum at a=

τ2 .

To guarantee that a 1, the companies should select a

=

min

τ2

; 1 provided that

they cover fixed costs. We summarize this in the next theorem.

Theorem 2. If a maximizes joint profits then

a=

1 τ

2;

τ

2 τ

2;

(19)

and

p

=

q

=

3r τ

+

4 τ

2;

r 2 τ 2:

(20)

Consequently, when τ 2 then a= 1 and the access fee coincides with the local

price. If the rate of substitutability is large, consumers will be charged a small

price (reflecting strong competition) and each company will be treated as any

other customer by the other company. If τ 2 (reflecting low substitutability), the

access fee is τ p

2 and this is smaller than the local price. Therefore, in equilibrium

the local price is r 2 , which is the monopoly price.

In this paper the attributes of the networks (other than the prices) are exoge-nously given. These attributes determine the value of σ and therefore τ. One

could add another stage to the game, for instance before a is determined. In this

stage the companies compete in attributes and their selections determine the value

of τ. Then Theorem 1 determines the equilibrium prices in terms of a and these

attributes. The equilibrium choice of attributes will be a function of their costs. If

the costs of different attributes are quite similar and a is selected to maximize joint

profits, then they will choose their attributes so that τ will be sufficiently small to

guarantee monopoly local prices (See Theorem 2). This will eliminate the effec-

tiveness of the price competition and will justify government intervention. An

extreme case is when government sets a to maximize total social surplus, SS , sub-

ject to the constraint that the companies cover fixed costs. The social surplus is

the sum of the total industry profits and the total consumer surplus, CS , where

CS =

N

4s

r

p

2:

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It is easy to verify that in equilibrium

SS =

Nr 2

4s τ +

4

2

τ+

2

a

τ+

6+

a

2F :

This function is a decreasing function of a for 0

a

1. Hence the social surplus

is maximized for the lowest level of a which still covers fixed costs. Observe that

if a=

0 then p

=

q

=

2r τ

+ 4 and

N

τ +

4

2s

τ + 2

r 2F

should hold to cover fixed costs. Hence fixed costs may not be covered for large

τ. This suggests that for a competitive industry the ”Bill and Keep” policy may jeopardize the viability of the industry.

4 Conclusion

We have analyzed the competition between two network companies which choose

access charges using RPACR. The most important result is that an equilibrium

in local prices always exists. The equilibrium prices exibit desirable properties

when the services of the networks are close substitutes and we believe that this

will be the case in a competitive telecommunications industry. The imposition of

RPACR can be viewed as a mild regulatory policy. One important advantage isthat implementation of RPACR does not require information about the industry

parameters. Since RPACR is responsive to future changes of the determinants

of the market, it provides flexibility for the companies to react to changes in the

environement and in strategies. For low differentiation between the networks, the

equilibirium local prices are low, provided that both companies cover their costs.

If one believes that this industry will not be as competitive and the companies will

be able to differentiate themselves considerably, then RPACR may not be useful.

It may serve as a collusion device for the companies to induce monopoly prices.

Further regulations should then be imposed to prevent such a case.

References

Anderson, S., A. de Palma, J.F. Thisse(1992): “ Discrete Choice Theory of Prod-

uct Differentiation”, MIT Press.

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Armstrong, M., C. Doyle, J. Vickers(1995):“ The Access Pricing Problem: A

Synthesis”, mimeo.

Baumol, W. J., G. Sidak(1994): “The Pricing of Inputs Sold to Competitors”, Yale

Journal of Regulation, vol.11, no.1, pp.171-202.

Carter, M., J. Wright(1996): “ Interconnection in Network Industries”, mimeo.

Domencich,T., D. McFadden(1975): “Urban Travel Demand”, North-Holland,

Amsterdam.

Economides, N. S., L.J. White(1995): “ Access and Interconnection Pricing: How

Efficient is the ”Efficient Component Pricing Rule”?”, Antitrust Bulletin,

forthcoming1 .

Economides, N. S., G. Woroch(1992):“ Benefits and Pitfalls of Network Intercon-

nection”, Stern School of Bussiness, Working Paper .

Laffont, J-J., J.Tirole (1996):“Network Competition: I. Overview and Nondis-

criminatory Pricing ”, mimeo, 38.

Laffont, J-J., J. Tirole(1994a) :“Creating Competition Through Interconnection:

Theory and Practice”, mimeo.

Laffont, J-J., J. Tirole(1994b) :“Access Pricing and Competition”, European Eco-

nomic Review, vol. 38, pp.1673-1710.

Muench, T(1988): “Quantum Agglomoration Formation During Growth in a

Combined Economic/Gravity Model”, Journal of Urban Economics, vol. 23,

pp. 199-214.

Noam, E.M.(1994):“ Interconnecting the Network of Networks”, AEI Working

Paper .

Vogelsang, I., B.M. Mitchell(1994):“ Telecommunications Competition: The Last

10 Miles”, AEI Working Paper .

Telecommunications Act of 1996, US Congress.

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Appendix

Proof of Theorem 1. Let α=

r

p and β=

r

q. Then the profit functions of

the two companies are

Π1

α;

β =

N

2s

α x

α x+ β x

α

r

α +

a

β

α

β x

α x+ β x

F ;

and

Π2

α;

β =

N

2s

β x

α x+ β x

β

r

β +

a

α

β

α x

α x+ β x

F :

Let

p

=

q

=

2 +

a

r

τ+

4:

Then

α

=

τ+

2

a

r

τ +

4=

β

:

We have to prove that Π1 α; β

is maximized for α = α and that Π2 α

; β is

maximized for β = β . By symmetry, it is enough to show this for Π1 α; β

.

Since Π1 α; β is continuous in the parameter τ, it is sufficient to prove our claim

for positive rational values of τ. Let τ=

mn

, where m

0, n

0 are integers. Let

t α

1n and s

β1n

:

Then t

= α

1n

=

s , where

s

= α

1n

=

m+ 2n

an

m+

4 r

1n

; (21)

and

Π1

t ;

s

N

2s

t m

t m+

sm

t n

r

t n +

ar

sn

t n

sm

t m+

sm

F :

Since t n is an increasing function of t , it is sufficient to prove that Π1

t ;

s

is

maximized for t

=

α

1

n when s

=

β

1

n .It can be easily verified that

∂Π1

∂t =

t m

t

t m+

sm

3P

t ;

s ;

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where the polynomial P

t ;

s is given by

P

t ;

s =

msmt n

r

t n

t m+

sm +

armsm

sn

t n

sm

t m

+

rnt n

2nt 2n

t m+

sm

2

arnt nsm

t m+

sm :

(22)

After rearranging the terms we have

P

t ;

s =

s2m

m+

2n

t 2n+

rm

arm+

rn

arn

t n+

armsn

sm

m+

4n

t m+

2n+

rm+

arm+

2rn

arn

t m+ n

armt nsn

2nt 2m+

2n+

rnt 2m+

n:

(23)

It is easy to verify that t =

t

=

s is the root of the polynomial P

t ;

s

. Therefore

P

t ;

s

=

t

s

g

t ;

s

;

(24)

for a certain polynomial g

t ;

s

. It can be verified that

g

t ;

s =

2nn 1

∑ j = 0

s jt 2m+

2n j

1+

n

r

2sn

2m 1

∑ j = 0

s jt 2m+ n j

1

wsmn 1

∑ j = 0

s jt 2n j

1+

y

wsn+

n

r

2sn

s2mn 1

∑ j = 0

s jt n j

1

vsm

n 1

∑ j = 0s

j

t m+

2n j

1+

z

vsn

n 1

∑ j = 0s

j

t m+ n j

1

+

z

vsn

arm

sm+ nn 1

∑ j =

0

s jt m j

1;

(25)

where s=

s

=

m+ 2n anm

+

4n r

1n and where

w=

m+

2n;

v=

m+

4n;

y=

rm

arm+

rn

arn;

(26)

z=

rm+

arm+ 2rn

arn:

It can be easily checked that

z

vsn

arm=

0;

(27)

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and

y

ws

n

n

r

2s

n =

arm:

(28)

Lemma 1. let 0

t

r 1n . Then, for 0

t

r 1= n, g

t ;

s

0.

Proof. Let us sum up the geometric series appearing in (25) and apply (26), (27),

(28). We obtain

g

t ;

s =

2nt 2m+

2n

1 1

st

n

1

st

+

n

r

2sn

t 2m+ n

1 1

st

2m

1

st

m+

2n

smt 2n

1 1

st

n

1

st

arms2mt n

1 1

st

n

1

st

(29)

m+

4n

smt m+ 2n 1 1

s

t

n

1

st

+

armsmt m+ n 1 1

s

t

n

1

st

;

where s=

s . Let us first show that the sum of the last three terms of the right-hand

side of (29) is negative. Consider the case where t

s . Observe that

arms2mt n

1

armsmt m+ n

1;

for all s

t . Therefore the sum of the last three terms of (29) is negative. Consider

next the case where t

s . It is sufficient to show that

m+

4n

smt m+

2n

1 1

st

n

1

s

t

armsmt m+

n

1 1

st

n

1

s

t

:

Therefore, it is sufficient to show that

m+ 4n

sn

arm 0: (30)

It is easy to check that (30) holds for s=

s . Consequently, the sum of the last

three terms of (29) is negative for every 0

t

r 1= n, and s=

s . Returning to

(29), we are left to show that when s=

s

g

t ;

s

2nt 2m+

2n

1 1

st

n

1

st

+

n

r

2sn

t 2m+ n

1

1

st

2m

1

st

m+

2n

smt 2n 1 1

s

t

n

1

st

0:

Let

h

t ;

s =

2t 2m

2s2m+

r

2sn

t 2m

s2m

t n

sn:

(31)

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It is easy to check that

g t ; s n

t

n

s

n

t

s

h t ; s :

Therefore, it is sufficient to show that h

t ;

s

0 for all 0

t

r 1= n and for s=

s .

To this end, we need the following lemma.

Lemma 2. Let

k

t ;

s =

t 2m

s2m

t n

sn:

Then, ∂k ∂t

t ;

s

0 if and only if

2m

n

t

s

0.

Proof. It can be easily verified that

∂k ∂t

t ; s = 2mt 2m 1

t n

sn

n2mt 2m n

t 2m

s2m

t n

sn 1

:

By the mean value theorem

t 2m

s2m

t n

sn=

2m

n c2m n

; (32)

where min

t ;

s

c

max

t ;

s . Therefore

∂k

∂t

t ;

s =

2mt 2m 1

t n

sn

h

c

t

2m n 1

i

;

and it is now easy to verify that the condition of Lemma 2 holds.

We will use Lemma 2 to prove that h

t ;

s 0 for every 0

t

r 1= n and s=

s .

First observe that by (31)

h

t ;

s

r

2sn

t 2m

s2m

t n

sn

2s2m

:

(33)

Consider the following four cases.

Case 1. 2m

n 0 and t

s .

By Lemma 2,

∂k

∂t

t ;

s

0. Hence, it is sufficient to show that the right-hand sideof (33) is negative at t =

s . Indeed for s=

s

h

s;

s

2m

n

r

2sn

s2m

n

2s2m=

s2m

2m

n

r

2sn

sn

2

:

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Using (21) , we just need to show that

2m

n

2an

m

m+ 2n

an

2 0; (34)

holds for all m and n such that 2m

n

0. It is sufficient to show that this is true

for a=

1 since the left-hand side of (34) is increasing in a. The last inequality

holds if and only if

m2+

n2

mn:

Clearly this is true for all m;

n

0; hence h

t ;

s

is negative for t

s .

Case 2. 2m

n

0 and t

s .

Again by Lemma 2, ∂k ∂t

t ;

s

0. Thus, we need to show that the right-hand side

of (33) is negative at t = 0. Indeed for s = s

h

0;

s

r

2sn

s2m

n

2s2m

s2m

r

2sn

sn

2

:

(35)

The right hand side of (35) is negative at s=

s if and only if

2an

m

m+ 2n

an 2 0;

for all m and n such that 2m

n 0. This is certainly true for a

= 1 and for all

m;

n

0.

Case 3. 2m n 0 and t s

.Since ∂k

∂t

t ;

s

0, we need to show that the right-hand side (33) is negative at

t =

s . For s=

s

h

s;

s

2m

n

r

2sn

s2m

n

2s2m=

s2m

2m

n

r

2sn

sn

2

;

and this was shown to be negative in Case 1.

Case 4. 2m

n

0 and t

s . In this case, by (31)

h

t ;

s

r

2sn

t 2m

s2m

t n

sn

2t 2m

:

(36)

By (32), for all s

t there exists c, s

c

t , such that

t 2m

s2m

t n

sn=

2m

n c2m n

:

(37)

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If 2m

n 0, c2m

n increases in c. Therefore, whenever s

t ,

c2m n

t 2m n

t 2m

sn:

Then by (36) and (37) we have

h

t ;

s

r 2sn

2m

n t 2ms n

2s2m=

t 2m

2m

n

r 2sn

sn 2

: (38)

The right-hand side of (38) is negative if and only if

2m

n

r

2sn

sn

2

0;

holds for s=

s . But in Case 1, it was shown that this condition is satisfied. This

completes the proof of Lemma 1.

By Lemma 1 and (24), P

t ;

s

0 whenever 0

t

s and P

t ;

s

0 whenever

s

t

r 1= n. Therefore, t =

s is the unique maximizer of Π1

t ;

s

and hence α

is the unique maximizer of Π1 α; β

. This completes the proof of Theorem 1.

19

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9611001